Pseudo-symplectic Runge-Kutta methods

Apart from specific methods amenable to specific problems, symplectic Runge-Kutta methods are necessarily implicit. The aim of this paper is to construct explicit Runge-Kutta methods which mimic symplectic ones as far as the linear growth of the global error is concerned. Such method of order p have to be pseudo-symplectic of pseudo-symplecticness order 2p, i.e. to preserve the symplectic form to within O(h(2p))-terms. Pseudo-symplecticness conditions are then derived and the effective construction of methods discussed. Finally, the performances of the new methods are illustrated on several test problems.